Tenney–Euclidean tuning: Difference between revisions

If we put the weighted Euclidean metric on tuning space, leading to TE tuning space in weighted coordinates, it is easy to find the nearest point in the subspace to the [[JIP]] {{val| 1 1 … 1 }}, and this closest point will define a [[tuning map]] which is called TE tuning, a tuning which has been extensively studied by [[Graham Breed]]. We may also keep unweighted coordinates and use the TE norm on tuning space; in these coordinates the JI point is {{val| 1 log₂3 … log₂''p'' }}. The two approaches are equivalent.  

There are a number of methods to solve least squares problems. One common way is to use the [[Moore–Penrose pseudoinverse]].

The Moore–Penrose pseudoinverse, denoted ''A''{{+}}, is a generalization of the inverse matrix with which it shares a lot of properties. In this method, the (not necessarily independent) TE generator map ''G'', which correspond to the rows of ''V'' are given by

Applying the val list to the generators, The TE tuning map is given by

We may also obtain the TE tuning from a [[projection matrix]]. {{nowrap| ''P''_''W'' {{=}} {{subsup|''V''|''W''|+}}''V''_''W'' }} is the orthogonal projection matrix that maps onto the space spanned by the rows of ''V''_''W''. This space corresponds to the temperament, and so does ''P''_''W''. However, ''P''_''W'' is independent of how the temperament is defined; it does not depend on whether the vals are linearly independent, how many of them there are, or whether [[contorsion]] has been removed. The weighted tuning map ''T''_''W'' = ''TW'' giving the weighted tuning of each prime number is found by multiplying by the JIP: {{nowrap| ''T''_''W'' {{=}} ''J''_''W''''P''_''W'' {{=}} ''J''_''W''{{subsup|''V''|''W''|+}}''V''_''W'' }}, which is the nearest point in the subspace corresponding to the temperament to ''J''_''W'', so that {{nowrap| ''T'' {{=}} ''J''_''W''''P''_''W''''W''{{inv}} {{=}} ''J''_''W''{{subsup|''V''|''W''|+}}''V'' }}.  

The weighting matrix ''W'' is given by

That simplifies the list of Frobenius generators, ''G'', to

where ''J'' is the non-weighted JIP and ''V'' is the nonweighted mapping.  

The Frobenius tuning map, ''T'', becomes

However, the main value of unweighted vals is that the pseudoinverse and projection matrix have rational entries, so that the rows of the matrix are [[fractional monzos]]. The Frobenius projection matrix therefore defines a completely canonical object not depending on any arbitrary definition (e.g. how Hermite normal form or LLL reduction is specifically defined) which corresponds one-to-one with temperaments, and which does not depend on whether the monzos or vals from which it is computed are [[Mathematical theory of saturation|saturated]]. It may be found starting either from a set of vals or a set of commas, since if ''Q'' is the projection matrix found by treating monzos in the same way as vals, {{nowrap|''P'' {{=}} ''I'' − ''Q''}} is the same projection matrix as would be found if starting from a set of vals defining the same temperament.

In D&D's guide, the systematic name for Frobenius tuning is ''minimax-E-copfr-S''.

'''Benedetti/Wilson–Euclidean tuning''' ('''BE tuning''') adopts the Benedetti or Wilson weight in place of Tenney weight, based on the dual norm of [[Wilson height]]. For {{nowrap|''Q'' {{=}} {{val| 2 3 5 … }}}}, the weighting matrix has the form  

In D&D's guide, the systematic name for BE tuning is ''minimax-E-sopfr-S''.

Consider [[pajara]], the 7-limit temperament tempering out 50/49 and 64/63. Its mapping ''V'' is

V = \begin{bmatrix}

2 & 0 & 11 & 12 \\

0 & 1 & -2 & -2

\end{bmatrix}

This can be found by taking two equal temperaments supporting pajara (e.g. [[12edo]] and [[22edo]]), defining a 2×4 matrix formed by the vals and then canonicalizing it. In weighted coordinates this would be

V_W =  

\begin{bmatrix}

0.000000 & 0.630930 & -0.861353 & -0.712414

\end{bmatrix}

Then the pseudoinverse ''V''{{+}} is approximately

V_W^+ =

\begin{bmatrix}

0.085876 & -0.060988

\end{bmatrix}

so we obtain the generator tuning map {{nowrap|''G'' {{=}} ''J''_''W''{{subsup|''V''|''W''|+}} }}:  

and the tuning map {{nowrap| ''T'' {{=}} ''GV'' }} which shows the value each prime harmonic is tuned to:  

The results are in octaves. Multiply them by 1200 to obtain the tuning in cents.  

If we use unweighted coordinates we get the Frobenius projection matrix instead, whose rows are [[fractional monzos]]. For instance, the unweighted pseudoinverse {{subsup|''V''|12|+}} of the 5-limit val ''V''₁₂ for 12 equal is the column matrix {{subsup|''V''|12|T}}/1289; that is, the 1×3 matrix with column {{monzo| 12/1289 19/1289 28/1289 }}. Then {{subsup|''V''|12|+}}''V''₁₂ is the 3×3 Frobenius projection matrix ''P''_F:

P_\text{F} = \frac{1}{1289}

\begin{bmatrix}

336 & 532 & 784

\end{bmatrix}

Multiplying {{val| 12 19 28 }} by ''P''_F gives {{val| 12 19 28 }} again. Multiplying the monzos for 81/80, 128/125, 648/625 etc. gives the zero monzo, corresponding to a unison. Multiplying the val for 5-limit 19 equal, {{val| 19 30 44 }}, by ''P''_F gives {{val| 24360 38570 56840 }}/1289, which is approximately the 19 equal val. Multiplying the 5-limit monzo for 3/2, which is {{monzo| -1 1 0 }}; times ''P''_F gives the fractional monzo corresponding to (2⁸⁴ 3¹³³ 5¹⁹⁶)^1/1289, which equates to 698.121 cents, the tempering of 3/2 in Frobenius tuning for 5-limit 12et, the tuning with octave defined by the top row of ''P''_F, which is to say by {{monzo| 1 0 0 }}''P''_F, of 1196.778 cents.

We can do the same thing with a matrix ''V'' for pajara; then ''P''_F, the Frobenius projection matrix, is

P_\text{F} = V^+V = \frac{1}{305}

\begin{bmatrix}

32 & 14 & 148 & 64

\end{bmatrix}

This sends monzos for 50/49, 64/63 etc. to the unison monzo, and vals for 10et, 12et, and 22et to themselves.